To analyze the expression [tex]\(4x^2yz + 3x^2y^2 + 16xy - 7\)[/tex], we evaluate the following:
### 1. Prime Factors in [tex]\(3x^2y^2\)[/tex]:
Examining the term [tex]\(3x^2y^2\)[/tex], the prime factors are components that are prime numbers:
- The coefficient 3 is a prime number.
- [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex] are variables and do not count as prime numbers.
Therefore, there is 1 prime factor in the term [tex]\(3x^2y^2\)[/tex] (which is 3).
### 2. Number of Terms:
To determine the number of terms in the expression [tex]\(4x^2yz + 3x^2y^2 + 16xy - 7\)[/tex]:
- Identify the terms by observing how many distinct summands are separated by plus (+) or minus (-) signs.
- The terms are: [tex]\(4x^2yz\)[/tex], [tex]\(3x^2y^2\)[/tex], [tex]\(16xy\)[/tex], and [tex]\(-7\)[/tex].
Therefore, there are 4 terms in the expression.
### 3. Number of Coefficients:
Coefficients are the numerical factors in each term:
- In [tex]\(4x^2yz\)[/tex], the coefficient is 4.
- In [tex]\(3x^2y^2\)[/tex], the coefficient is 3.
- In [tex]\(16xy\)[/tex], the coefficient is 16.
- In [tex]\(-7\)[/tex], the coefficient is [tex]\(-7\)[/tex].
For counting coefficients, considering if any of them repeat is unnecessary since we are interested in the total count.
Therefore, there are 4 coefficients in the expression.
### 4. Number of Constants:
A constant term is a term without any variables. In our expression:
- [tex]\(-7\)[/tex] is the only term without variables.
Thus, there is 1 constant in the expression.
### Summary:
1. The number of prime factors in [tex]\(3x^2y^2\)[/tex] is [tex]\( \boxed{1} \)[/tex].
2. The number of terms is [tex]\( \boxed{4} \)[/tex].
3. The number of coefficients is [tex]\( \boxed{4} \)[/tex].
4. The number of constants is [tex]\( \boxed{1} \)[/tex].
